A Scala object cannot inherit from another Scala object so the obvious way is not possible.
If you can modify the original object then create a class that implements all the functionality and make the original object inherit from that class. Your new object can then inherit from the same class and override the methods that you want to change. However this will create two copies of any values in the base class, so it is not suitable for an object that contains a lot of data or does any one-off initialisation.
If you cannot modify the original object then you will have to copy all the methods of the first object in your new object. vals can be copied directly. defs can be copied using eta expansion:
def v = Original.v // This is a simple value
def f = Original.f _ // This is a function value
Using def rather than val here will avoid storing multiple copies of the original values and will prevent lazy values from being computed until they are needed.
Using eta expansion will make f a function value rather than a method which may or may not be a problem depending on how it is used. If you require f to be a method then you will have to duplicate the function signature and call the original f:
def f(i: Int) = Original.f(i) // This is a method